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[b]1. The problem statement, all variables and given/known data[/b]

The average growth rate of the population of a certain city is 7.5% per year. The city's population is now 22,750 people. What is the expected population in 10 years?

[b]2. Relevant equations[/b]

I was always taught this formula for exponential growth

N(t) = N_0 e^(kt)
N = population
N_0 = population at t(0)
e = 2.7...
k = some positive constant
t = time

Here's what my teacher wrote on my paper for the formula

f(x) = C(1 + r)^x
f(x) = population
C = initial population
r = growth rate

[b]3. The attempt at a solution[/b]

no i don't understand how to do this exactly because I don't know what to use for the constant k

so i used the second one

22750 (1 + .075)^10 = 46888.4680

now what I don't udnerstand is that this really makes no sense at all becasue if I wanted to fidn the population at 10 minutes or ten centuries and just plugged in 10 into the equation with no units at all I would get the same exact answer. Can youp please tell me how to go about reasoning this out... THANKS!

  • PreCalc -

    When your teacher wrote ...

    f(x) = C(1 + r)^x
    f(x) = population
    C = initial population
    r = growth rate

    he/she should have also defined x to be the annual rate.
    so when you replaced x with 10 it was understood that it was years, since the r was the rate per year

    If you wanted to find out for a time other than years, you would have to change t to that fraction of a year.
    e.g. if you only wanted it for 10 months, t = 10/12 or .83333..
    if you wanted 10 minutes you would have to find the number of minutes in a year
    which is 365*24*60 = 525600
    so your exponent for 10 minutes would not be 10 but

    the second equation is probably the easier to use for these types of questions.

    you could use the first one

    N(t) = N0 e^(kt)
    here we have to find the value of k first by using one set of data given, that is
    when t = 1,
    22750(1.075) = 22750e^1k)
    e^k = 1.075
    k = ln 1.075 = .07232

    so N(t) = 22750e^.07232t
    so when t=10
    N(10) = 22750(e^(.07232)(10))
    = 46888.46804 exactly the same as obtained with the other formula.

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